ñòð. 6 |

+ Prob CFt=1 will be case 2 Â· CFt=1 cash ï¬‚ow in case 2

+ Prob CFt=1 will be case 3 Â· CFt=1 cash ï¬‚ow in case 3 .

This expected return of $206.80 is less than the $210 that the government promises. Put

diï¬€erently, if I promise you a rate of return of 5%,

$210 âˆ’ $200

Promised(Ëœt=0,1 ) = = 5.00%

r

$200

(5.11)

Ëœ

Promised(Ct=1 ) âˆ’ CFt=0

Promised(Ëœt=0,1 ) =

r ,

CFt=0

your expected rate of return is only

$206.80 âˆ’ $200

E (Ëœt=0,t=1 ) = = 3.40%

r

$200

(5.12)

Ëœ

E (Ct=1 ) âˆ’ CFt=0

E (Ëœt=0,t=1 ) =

r ,

CFt=0

which is less than the 5% interest rate that Uncle Sam promisesâ€”and surely delivers.

You need to determine how much I have to promise you to â€œbreak even,â€ so that you expect to Determine how much

more interest promise

end up with the same $210 that you could receive from Uncle Sam. In Table 5.1, we computed

you need to break even.

the expected payoï¬€ as the probability-weighted average payoï¬€. You want this payoï¬€ to be not

ï¬le=uncertainty.tex: LP

90 Chapter 5. Uncertainty, Default, and Risk.

$206.80, but the $210 that you can get if you put your money in government bonds. So, you

now solve for an amount x that you want to receive if I have money,

Ëœ

E (Ct=1 ) = Â· x

98%

+ Â·

1% $100

+ Â· = $210.00

1% $0

(5.13)

Ëœ

E (Ct=1 ) = Prob CFt=1 will be case 1 Â· CFt=1 cash ï¬‚ow in case 1

+ Prob CFt=1 will be case 2 Â· CFt=1 cash ï¬‚ow in case 2

+ Prob CFt=1 will be case 3 Â· CFt=1 cash ï¬‚ow in case 3 .

The solution is that if I promise you x = $213.27, you will expect to earn the same 5% interest

rate that you can earn in Treasury bonds. Table 5.2 conï¬rms that a promise of $213.27 for a

cash investment of $200, which is a promised interest rate of

$213.27 âˆ’ $200

Promised(Ëœt=0,1 ) = = 6.63%

r

$200

(5.14)

Ëœ

Promised(Ct=1 ) âˆ’ CFt=0

Promised(Ëœt=0,1 ) =

r ,

CFt=0

and provides an expected interest rate of

(5.15)

E (Ëœt=0,1 ) = 98% Â· (+6.63%) + 1% Â· (âˆ’50%) + 1% Â· (âˆ’100%) = 5% .

r

Table 5.2. Risky Payoï¬€ Table: 6.63% Promised Interest Rate

How Likely

Receive Cash which is a

Flow CFt=1 return of (Probability)

$213.27 +6.63% 98% of the time

$100.00 â€“50.00% 1% of the time

$0.00 â€“100.00% 1% of the time

$210.00 +5.00% in expectation

The diï¬€erence of 1.63% between the promised (or quoted) interest rate of 6.63% and the ex-

The difference between

the promised and pected interest rate of 5% is the default premiumâ€”it is the extra interest rate that is caused by

expected interest rate is

the default risk. Of course, you only receive this 6.63% if everything goes perfectly. In our

the default premium.

perfect world with risk-neutral investors,

= +

6.63% 5% 1.63%

(5.16)

â€œPromised Interest Rateâ€ = â€œTime Premiumâ€ + â€œDefault Premiumâ€ .

Important: Except for 100% safe bonds (Treasuries), the promised (or quoted)

rate of return is higher than the expected rate of return. Never confuse the higher

promised rate for the lower expected rate.

Financial securities and information providers rarely, if ever, provide expected

rates of return.

ï¬le=uncertainty.tex: RP

91

Section 5Â·2. Interest Rates and Credit Risk (Default Risk).

On average, the expected rate of return is the expected time premium plus the expected default In a risk-neutral world,

all securities have the

premium. Because the expected default premium is zero on average,

same exp. rate of return.

E Rate of Return = E Time Premium + E Realized Default Premium

(5.17)

= E Time Premium + .

0

If you want to work this out, you can compute the expected realized default premium as follows:

you will receive (6.63% âˆ’ 5% = 1.63%) in 98% of all cases; âˆ’50% âˆ’ 5% = âˆ’55% in 1% of all cases

(note that you lose the time-premium); and âˆ’100% âˆ’ 5% = âˆ’105% in the remaining 1% of all

cases (i.e., you lose not only all your money, but also the time-premium). Therefore,

(5.18)

E Realized Default Premium = 98% Â· (+1.63%) + 1% Â· (âˆ’55%) + 1% Â· (âˆ’105%) = 0% .

Solve Now!

Q 5.6 Recompute the example in Table 5.2 assuming that the probability of receiving full pay-

ment of $210 is only 95%, the probability of receiving $100 is 1%, and the probability of receiving

absolutely no payment is 4%.

(a) At the promised interest rate of 5%, what is the expected interest rate?

(b) What interest rate is required as a promise to ensure an expected interest rate of 5%?

5Â·2.C. Preview: Risk-Averse Investors Have Demanded Higher Expected Rates

We have assumed that investors are risk-neutralâ€”indiï¬€erent between two loans that have the In addition to the

default premium, in real

same expected rate of return. As we have already mentioned, in the real world, risk-averse

life, investors also

investors would demand and expect to receive a little bit more for the risky loan. Would you demand a risk premium.

rather invest into a bond that is known to pay oï¬€ 5% (for example, a U.S. government bond), or

would you rather invest in a bond that is â€œmerelyâ€ expected to pay oï¬€ 5% (such as my 6.63%

bond)? Like most lenders, you are likely to be better oï¬€ if you know exactly how much you will

receive, rather than live with the uncertainty of my situation. Thus, as a risk-averse investor,

you would probably ask me not only for the higher promised interest rate of 6.63%, which only

gets you to an expected interest rate of 5%, but an even higher promise in order to get you more

than 6.63%. For example, you might demand 6.75%, in which case you would expect to earn

not just 5%, but a little more. The extra 12 basis points is called a risk premium, and it is an

interest component required above and beyond the time premium (i.e., what the U.S. Treasury

Department pays for use of money over time) and above and beyond the default premium (i.e.,

what the promised interest has to be for you to just expect to receive the same rate of return

as what the government oï¬€ers).

Recapping, we know that 5% is the time-value of money that you can earn in interest from the A more general

decomposition of rates

Treasury. You also know that 1.63% is the extra default premium that I must promise you, a

of return.

risk-neutral lender, to allow you to expect to earn 5%, given that repayment is not guaranteed.

Finally, if you are not risk-neutral but risk-averse, I may have to pay even more than 6.63%,

although we do not know exactly how much.

If you want, you could think of further interest decompositions. It could even be that the time- More intellectually

interesting, but

premium is itself determined by other factors (such as your preference between consuming

otherwise not too useful

today and consuming next year, the inï¬‚ation rate, taxes, or other issues, that we are brushing decompositions.

over). Then there would be a liquidity premium, an extra interest rate that a lender would

demand if the bond could not easily be soldâ€”resale is much easier with Treasury bonds.

ï¬le=uncertainty.tex: LP

92 Chapter 5. Uncertainty, Default, and Risk.

Important: When repayment is not certain, lenders demand a promised interest

rate that is higher than the expected interest rate by the default premium.

Promised Interest Rate

(5.19)

= Time Premium + Default Premium + Risk Premium .

The promised default premium is positive, but it is only paid when everything goes

well. The actually earned interest rate consists of the time premium, the realized

risk premium, and a (positive or negative) default realization.

Actual Interest Rate Earned

(5.20)

= Time Premium + Default Realization + Risk Premium .

The default realization could be more than negative enough to wipe out both the

time premium and the risk premium. But it is zero on average. Therefore,

Expected Interest Rate

(5.21)

= Time Premium + Expected Risk Premium .

The risk premium itself depends on such strange concepts as the correlation of loan default

Some real world

evidence. with the general economy and will be the subject of Part III of the book. However, we can preview

the relative importance of these components for you in the context of corporate bonds. (We

will look at risk categories of corporate bonds in more detail in the next chapter.) The highest-

quality bonds are called investment-grade. A typical such bond may promise about 6% per

annum, 150 to 200 basis points above the equivalent Treasury. The probability of default

would be smallâ€”less than 3% in total over a ten-year horizon (0.3% per annum). When an

investment-grade bond does default, it still returns about 75% of what it promised. For such

bonds, the risk premium would be smallâ€”a reasonable estimate would be that only about 10

to 20 basis points of the 200 basis point spread is the risk premium. The quoted interest rate

of 6% per annum therefore would reï¬‚ect ï¬rst the time premium, then the default premium,

and only then a small risk premium. (In fact, the liquidity premium would probably be more

important than the risk premium.) For low-quality corporate bonds, however, the risk premium

can be important. Ed Altman has been collecting corporate bond statistics since the 1970s. In

an average year, about 3.5% to 5.5% of low-grade corporate bonds defaulted. But in recessions,

the default rate shot up to 10% per year, and in booms it dropped to 1.5% per year. The average

value of a bond after default was only about 40 cents on the dollar, though it was as low 25

cents in recessions and as high as 50 cents in booms. Altman then computes that the most

risky corporate bonds promised a spread of about 5%/year above the 10-Year Treasury bond,

but ultimately delivered a spread of only about 2.2%/year. 280 points are therefore the default

premium. The remaining 220 basis points contain both the liquidity premium and the risk

premiumâ€”perhaps in roughly equal parts.

Solve Now!

Q 5.7 Return to the example in Table 5.2. Assume that the probability of receiving full payment

of $210 is only 95%, the probability of receiving $100 is 4%, and the probability of receiving

absolutely no payment is 1%. If the bond quotes a rate of return of 12%, what is the time premium,

the default premium and the risk premium?

ï¬le=uncertainty.tex: RP

93

Section 5Â·3. Uncertainty in Capital Budgeting, Debt, and Equity.

5Â·3. Uncertainty in Capital Budgeting, Debt, and Equity

We now turn to the problem of selecting projects under uncertainty. Your task is to compute

present values with imperfect knowledge about future outcomes. Your principal tool in this

task will be the payoï¬€ table (or state table), which assigns probabilities to the project value in

each possible future value-relevant scenario. For example, a ï¬‚oppy disk factory may depend

on computer sales (say, low, medium, or high), whether ï¬‚oppy disks have become obsolete

(yes or no), whether the economy is in a recession or expansion, and how much the oil price

(the major cost factor) will be. Creating the appropriate state table is the managerâ€™s taskâ€”

judging how the business will perform depending on the state of these most relevant variables.

Clearly, it is not an easy task even to think of what the key variables are, to determine the

probabilities under which these variables will take on one or another value. Assessing how

your own project will respond to them is an even harder taskâ€”but it is an inevitable one. If

you want to understand the value of your project, you must understand what the projectâ€™s key

value drivers are and how the project will respond to these value drivers. Fortunately, for many

projects, it is usually not necessary to describe possible outcomes in the most minute detailâ€”

just a dozen or so scenarios may be able to cover the most important information. Moreover,

these state tables will also allow you to explain what a loan (also called debt or leverage) and

levered ownership (also called levered equity) are, and how they diï¬€er.

5Â·3.A. Present Value With State-Contingent Payoï¬€ Tables

Almost all companies and projects are ï¬nanced with both debt and levered equity. We already Most projects are

ï¬nanced with a mix of

know what debt is. Levered equity is simply what accrues to the business owner after the debt

debt and equity.

is paid oï¬€. (In this chapter, we shall not make a distinction between ï¬nancial debt and other

obligations, e.g., tax obligations.) You already have an intuitive sense about this. If you own a

house with a mortgage, you really own the house only after you have made all debt payments.

If you have student loans, you yourself are the levered owner of your future income stream.

That is, you get to consume â€œyourâ€ residual income only after your liabilities (including your

non-ï¬nancial debt) are paid back. But what will the levered owner and the lender get if the

companyâ€™s projects fail, if the house collapses, or if your career takes a turn towards Rikers

Island? What is the appropriate compensation for the lender and the levered owner? The split

of net present value streams into loans (debt) and levered equity lies at the heart of ï¬nance.

We will illustrate this split through the hypothetical purchase of a building for which the fu- The example of this

section: A building in

ture value is uncertain. This building is peculiar, though: it has a 20% chance that it will be

Tornado Alley can have

destroyed, say by a tornado, by next year. In this case, its value will only be the landâ€”say, one of two possible

$20,000. Otherwise, with 80% probability, the building will be worth $100,000. Naturally, the future values.

$100,000 market value next year would itself be the result of many factorsâ€”it could include

any products that have been produced inside the building, real-estate value appreciation, as

well as a capitalized value that takes into account that a tornado might strike in subsequent

years.

Table 5.3. Building Payoï¬€ Table

Event Probability Value

Tornado 20% $20,000

Sunshine 80% $100,000

Expected Future Value $84,000

ï¬le=uncertainty.tex: LP

94 Chapter 5. Uncertainty, Default, and Risk.

The Expected Building Value

Table 5.3 shows the payoï¬€ table for full building ownership. The expected future building

To obtain the expected

future cash value of the value of $84,000 was computed as

building, multiply each

(possible) outcome by its

probability. E (Valuet=1 ) = Â·

20% $20, 000

+ Â· = $84, 000

80% $100, 000

(5.22)

= Prob( Tornado ) Â· (Value if Tornadot=1 )

+ Prob( Sunshine ) Â· (Value if Sunshinet=1 ) .

Now, assume that the appropriate expected rate of return for a project of type â€œbuildingâ€ with

Then discount back the

expected cash value this type of riskiness and with one-year maturity is 10%. (This 10% discount rate is provided by

using the appropriate

demand and supply in the ï¬nancial markets and known.) Your goal is to determine the present

cost of capital.

valueâ€”the appropriate correct priceâ€”for the building today.

There are two methods to arrive at the present value of the buildingâ€”and they are almost

Under uncertainty, use

NPV on expected (rather identical to what we have done earlier. We only need to replace the known value with the

than actual, known) cash

expected value, and the known future rate of return with an expected rate of return. Now, the

ï¬‚ows, and use the

ï¬rst PV method is to compute the expected value of the building next period and to discount

appropriate expected

(rather than actual,

it at the cost of capital, here 10 percent,

known) rates of return.

The NPV principles

$84, 000

remain untouched. PVt=0 = â‰ˆ $76, 363.64

1 + 10%

(5.23)

E (Valuet=1 )

= .

1 + E (rt=0,1 )

Table 5.4. Building Payoï¬€ Table, Augmented

â‡’

Event Probability Value Discount Factor PV

â‡’

1/(1+10%)

Tornado 20% $20,000 $18,181.82

â‡’

1/(1+10%)

Sunshine 80% $100,000 $90,909.09

The second method is to compute the discounted state-contingent value of the building, and

Taking expectations and

discounting can be done then take expected values. To do this, augment Table 5.3. Table 5.4 shows that if the tornado

in any order.

strikes, the present value is $18,181.82. If the sun shines, the present value is $90,909.10.

Thus, the expected value of the building can also be computed

PVt=0 = Â·

20% $18, 181.82

+ Â· â‰ˆ $76, 363.64

80% $90, 909.09

(5.24)

= Prob( Tornado ) Â· (PV of Building if Tornado)

+ Prob( Sunshine ) Â· (PV of Building if Sunshine) .

Both methods lead to the same result: you can either ï¬rst compute the expected value next year

(20% Â· $20, 000 + 80% Â· $100, 000 = $84, 000), and then discount this expected value of $84,000

to $76,363.34; or you can ï¬rst discount all possible future outcomes ($20,000 to $18,181.82;

and $100,000 to $90,909.09), and then compute the expected value of the discounted values

(20% Â· $18, 181.82 + 80% Â· $90, 909.09 = $76, 363.34.)

ï¬le=uncertainty.tex: RP

95

Section 5Â·3. Uncertainty in Capital Budgeting, Debt, and Equity.

Important: Under uncertainty, in the NPV formula,

â€¢ known future cash ï¬‚ows are replaced by expected discounted cash ï¬‚ows, and

â€¢ known appropriate rates of return are replaced by appropriate expected

rates of return.

You can ï¬rst do the discounting and then take expectations, or vice-versa.

The State-Dependent Rates of Return

What would the rates of return be in both states, and what would the overall expected rate of The state-contingent

rates of return can also

return be? If you have bought the building for $76,363.64, and no tornado strikes, your actual

be probability-weighted

rate of return (abbreviated rt=0,1 ) will be to arrive at the average

(expected) rate of

$100, 000 âˆ’ $76, 363.64 return.

rt=0,1 = â‰ˆ +30.95% . (5.25)

if Sunshine:

$76, 363.64

If the tornado does strike, your rate of return will be

$20, 000 âˆ’ $76, 363.64

rt=0,1 = â‰ˆ âˆ’73.81% . (5.26)

if Tornado:

$76, 363.64

Therefore, your expected rate of return is

E (Ëœt=0,1 ) = Â·

r (âˆ’73.81%)

20%

+ Â· = 10.00%

(+30.95%)

80%

(5.27)

E (Ëœt=0,1 ) = Prob( Tornado ) Â· (rt=0,1 if Tornado)

r

+ Prob( Sunshine ) Â· (rt=0,1 if Sunshine) .

The probability state-weighted rates of return add up to the expected overall rate of return.

This is as it should be: after all, we derived the proper price of the building today using a 10%

expected rate of return.

Solve Now!

Q 5.8 What changes have to be made to the NPV formula to handle an uncertain future?

Q 5.9 Under risk-neutrality, a factory can be worth $500,000 or $1,000,000 in two years, de-

pending on product demand, each with equal probability. The appropriate cost of capital is 6%

per year. What is the present value of the factory?

Q 5.10 A new product may be a dud (20% probability), an average seller (70% probability) or

dynamite (10% probability). If it is a dud, the payoï¬€ will be $20,000; if it is an average seller, the

payoï¬€ will be $40,000; if it is dynamite, the payoï¬€ will be $80,000. What is the expected payoï¬€

of the project?

Q 5.11 (Continued.) The appropriate expected rate of return for such payoï¬€s is 8%. What is the

PV of the payoï¬€?

Q 5.12 (Continued.) If the project is purchased for the appropriate present value, what will be

the rates of return in each of the three outcomes?

Q 5.13 (Continued.) Conï¬rm the expected rate of return when computed from the individual

outcome speciï¬c rates of return.

ï¬le=uncertainty.tex: LP

96 Chapter 5. Uncertainty, Default, and Risk.

5Â·3.B. Splitting Project Payoï¬€s into Debt and Equity

We now know how to compute the NPV of state-contingent payoï¬€sâ€”our building paid oï¬€ dif-

State-contingent claims

have payoffs that ferently in the two states of nature. Thus, our building was a state-contingent claimâ€”its payoï¬€

depend on future states

depended on the outcome. But it is just one of many. Another state-contingent claim might

of nature.

promise to pay $1 if the sun shines and $25 if a tornado strikes. Using payoï¬€ tables, we can

work out the value of any state-contingent claimsâ€”and in particular the value of the two most

important state-contingent claims, debt and equity.

The Loan

We now assume you want to ï¬nance the building purchase of $76,363.64 with a mortgage of

Assume the building is

funded by a mortgagor $25,000. In eï¬€ect, the single project â€œbuildingâ€ is being turned into two diï¬€erent projects,

and a residual, levered

each of which can be owned by a diï¬€erent party. The ï¬rst project is the project â€œMortgage

building owner.

Lending.â€ The second project is the project â€œResidual Building Ownership,â€ i.e., ownership of

the building but bundled with the obligation to repay the mortgage. This â€œResidual Building

Ownershipâ€ investor will not receive a dime until after the debt has been satisï¬ed. Such residual

ownership is called the levered equity, or just the equity, or even the stock in the building, in

order to avoid calling it â€œwhatâ€™s-left-over-after-the-loans-have-been-paid-oï¬€.â€

What sort of interest rate would the creditor demand? To answer this question, we need to

The ï¬rst goal is to

determine the know what will happen if the building were to be condemned, because the mortgage value

appropriate promised

($25,000 today) will be larger than the value of the building if the tornado strikes ($20,000

interest rate on a

next year). We are assuming that the owner could walk away from it and the creditor could

â€œ$25,000 value todayâ€

mortgage loan on the

repossess the building, but not any of the borrowerâ€™s other assets. Such a mortgage loan is

building.

called a no-recourse loan. There is no recourse other than taking possession of the asset itself.

This arrangement is called limited liability. The building owner cannot lose more than the

money that he originally puts in. Limited liability is a mainstay of many ï¬nancial securities: for

example, if you purchase stock in a company in the stock market, you cannot be held liable for

more than your investment, regardless of how badly the company performs.

Table 5.5. Payoï¬€ to Mortgage Creditor, Providing $25,000 Today

Prob

Event Value Discount Factor

1/(1+10%)

Tornado 20% $20,000

1/(1+10%)

Sunshine 80% Promised

Anecdote:

The framers of the United States Constitution had the English bankruptcy system in mind when they included

the power to enact â€œuniform laws on the subject of bankruptciesâ€ in the Article I Powers of the legislative branch.

The ï¬rst United States bankruptcy law, passed in 1800, virtually copied the existing English law. United States

bankruptcy laws thus have their conceptual origins in English bankruptcy law prior to 1800. On both sides of

the Atlantic, however, much has changed since then.

Early English law had a distinctly pro-creditor orientation, and was noteworthy for its harsh treatment of de-

faulting debtors. Imprisonment for debt was the order of the day, from the time of the Statute of Merchants

in 1285, until Dickensâ€™ time in the mid-nineteenth century. The common law writs of capias authorized â€œbody

execution,â€ i.e., seizure of the body of the debtor, to be held until payment of the debt.

English law was not unique in its lack of solicitude for debtors. Historyâ€™s annals are replete with tales of

draconian treatment of debtors. Punishments inï¬‚icted upon debtors included forfeiture of all property, re-

linquishment of the consortium of a spouse, imprisonment, and death. In Rome, creditors were apparently

authorized to carve up the body of the debtor, although scholars debate the extent to which the letter of that

law was actually enforced.

Direct Source: Charles Jordan Tabb, 1995, â€œThe History of the Bankruptcy laws in the United States.â€

www.bankruptcyï¬nder.com/historyofbkinusa.html. (The original article contains many more juicy historical

tidbits.)

ï¬le=uncertainty.tex: RP

97

Section 5Â·3. Uncertainty in Capital Budgeting, Debt, and Equity.

To compute the PV for the project â€œMortgage Lending,â€ we return to the problem of setting Start with the Payoff

Table, and write down

an appropriate interest rate, given credit risk (from Section 5Â·2). Start with the payoï¬€ table in

payoffs to project

Table 5.5. The creditor receives the property worth $20,000 if the tornado strikes, or the full â€œMortgage Lending.â€

promised amount (to be determined) if the sun shines. To break even, the creditor must solve

for the payoï¬€ to be received if the sun will shine in exchange for lending $25,000 today. This

is the â€œquotedâ€ or â€œpromisedâ€ payoï¬€.

$20, 000

= Â·

$25, 000 20%

1 + 10%

Promise

+ Â·

80%

1 + 10%

(5.28)

Loan Valuet=0 = Prob( Tornado ) Â· (Loan PV if Tornado)

+ Prob( Sunshine ) Â· (Loan PV if Sunshine) .

Solving, the solution is a promise of

(1 + 10%) Â· $25, 000 âˆ’ 20% Â· $20, 000

Promise = = $29, 375

80%

(5.29)

[1 + E (r )] Â· Loan Value âˆ’ Prob(Tornado) Â· Value if Tornado

=

Prob(Sunshine)

in repayment, paid by the borrower only if the sun will shine.

With this promised payoï¬€ of $29,375 (if the sun will shine), the lenderâ€™s rate of return will be The state-contingent

rates of return in the

the promised rate of return:

tornado (â€œdefaultâ€)

state and in the sunshine

$29, 375 âˆ’ $25, 000 state can be probability

rt=0,1 = = +17.50% , (5.30)

if Sunshine:

$25, 000 weighted to arrive at the

expected rate of return.

The lender would not provide the mortgage at any lower promised interest rate. If the tornado

strikes, the owner walks away, and the lenderâ€™s rate of return will be

$20, 000 âˆ’ $25, 000

rt=0,1 = = âˆ’20.00% . (5.31)

if Tornado:

$25, 000

Therefore, the lenderâ€™s expected rate of return is

E (Ëœt=0,1 ) = Â·

r (âˆ’20.00%)

20%

+ Â· = 10.00%

(+17.50%)

80%

(5.32)

E (Ëœt=0,1 ) = Prob( Tornado ) Â· (rt=0,1 if Tornado)

r

+ Prob( Sunshine ) Â· (rt=0,1 if Sunshine) .

After all, in a risk-neutral environment, anyone investing for one year expects to earn an ex-

pected rate of return of 10%.

The Levered Equity

Our interest now turns towards proper compensation for youâ€”the expected payoï¬€s and ex- Now compute the

payoffs of the 60%

pected rate of return for you, the residual building owner. We already know the building

post-mortgage (i.e.,

is worth $76,363.64 today. We also already know how the lender must be compensated: to levered) ownership of

contribute $25,000 to the building price today, you must promise to pay the lender $29,375 the building. The

method is exactly the

next year. Thus, as residual building owner, you need to pay $51,363.64â€”presumably from

same.

personal savings. If the tornado strikes, these savings will be lost and the lender will repos-

sess the building. However, if the sun shines, the building will be worth $100,000 minus the

promised $29,375, or $70,625. The ownerâ€™s payoï¬€ table in Table 5.6 allows you to determine

ï¬le=uncertainty.tex: LP

98 Chapter 5. Uncertainty, Default, and Risk.

that the expected future levered building ownership payoï¬€ is 20%Â·$0+80%Â·$70, 625 = $56, 500.

Therefore, the present value of levered building ownership is

$0 $70, 625

PVt=0 = 20% Â· + 80% Â·

1 + 10% 1 + 10%

(5.33)

= Prob( Tornado ) Â· (PV if Tornado) + Prob( Sunshine ) Â· (PV if Sunshine)

= $51, 363.64 .

Table 5.6. Payoï¬€ To Levered Building (Equity) Owner

Prob

Event Value Discount Factor

1/(1+10%)

Tornado 20% $0.00

1/(1+10%)

Sunshine 80% $70,624.80

If the sun shines, the rate of return will be

Again, knowing the

state-contingent cash

$70, 624.80 âˆ’ $51, 363.63

ï¬‚ows permits computing

rt=0,1 = = +37.50% . (5.34)

if Sunshine:

state-contingent rates of

$51, 363.63

return and the expected

rate of return.

If the tornado strikes, the rate of return will be

$0 âˆ’ $51, 363.63

rt=0,1 = = âˆ’100.00% . (5.35)

if Tornado:

$51, 363.63

The expected rate of return of levered equity ownership, i.e., the building with the bundled

mortgage obligation, is

E (Ëœt=0,1 ) = Â·

r (âˆ’100.00%)

20%

+ Â· = 10.00%

(+37.50%)

80%

(5.36)

E (Ëœt=0,1 ) = Prob( Tornado ) Â· (rt=0,1 if Tornado)

r

+ Prob( Sunshine ) Â· (rt=0,1 if Sunshine) .

Reï¬‚ections On The Example: Payoï¬€ Tables

Payoï¬€ tables are fundamental tools to think about projects and ï¬nancial claims. You should

think about ï¬nancial claims in terms of payoï¬€ tables.

Important: Whenever possible, in the presence of uncertainty, write down a

payoï¬€ table to describe the probabilities of each possible event (â€œstateâ€) with its

state-contingent payoï¬€s.

Admittedly, this can sometimes be tedious, especially if there are many diï¬€erent possible states

(or even inï¬nitely many states, as in a bell-shaped normally distributed project outcomeâ€”but

you can usually approximate even the most continuous and complex outcomes fairly well with

no more than ten discrete possible outcomes), but they always work!

ï¬le=uncertainty.tex: RP

99

Section 5Â·3. Uncertainty in Capital Budgeting, Debt, and Equity.

Table 5.7. Payoï¬€ Table and Overall Values and Returns

Prob

Event Building Value Mortgage Value Levered Ownership

Tornado 20% $20,000 $20,000 $0

Sunshine 80% $100,000 $29,375 $70,625

Expected Valuet=1 $84,000 $27,500 $56,500

PVt=0 $76,364 $25,000 $51,364

E(rt=0,1 ) 10% 10% 10%

Table 5.7 shows how elegant such a table can be. It can describe everything we need in a There are three possible

investment

very concise manner: the state-contingent payoï¬€s, expected payoï¬€s, net present value, and

opportunities here. The

expected rates of return for your building scenario. Because owning the mortgage and the bank is just another

levered equity is the same as owning the full building, the last two columns must add up to the investor, with particular

payoff patterns.

values in the â€œbuilding valueâ€ column. You could decide to be any kind of investor: a creditor

(bank) who is loaning money in exchange for promised payment; a levered building owner who

is taking a â€œpiece left over after a loanâ€; or an unlevered building owner who is investing money

into an unlevered project. You might take the whole piece (that is, 100% of the claimâ€”all three

investments are just claims) or you might just invest, say $5, at the appropriate fair rates of

return that are due to investors in our perfect world, where everything can be purchased at or

sold at a fair price. (Further remaining funds can be raised elsewhere.)

Reï¬‚ections On The Example and Debt and Equity Risk

We have not covered risk yet, because we did not need to. In a risk-neutral world, all that

matters is the expected rate of return, not how uncertain you are about what you will receive.

Of course, we can assess risk even in a risk-neutral world, even if risk were to earn no extra

compensation (a risk premium).

So, which investment is most risky: full ownership, loan ownership, or levered ownership? Leveraging (mortgaging)

a project splits it into a

Figure 5.2 plots the histograms of the rates of return to each investment type. As the visual

safer loan and a riskier

shows, the loan is least risky, followed by the full ownership, followed by the levered ownership. levered ownership,

Your intuition should tell you that, by taking the mortgage, the medium-risky project â€œbuildingâ€ although everyone

expects to receive 10%

has been split into a more risky project â€œlevered buildingâ€ and a less risky project â€œmortgage.â€

on average.

The combined â€œfull building ownershipâ€ project therefore has an average risk.

It should not come as a surprise to learn that all investment projects expect to earn a 10% If everyone is

risk-neutral, everyone

rate of return. After all, 10% is the time-premium for investing money. Recall from Page 92

should expect to earn

that the expected rate of return (the cost of capital) consists only of a time-premium and a 10%.

risk premium. (The default premium is a component only of promised interest rates, not of

expected interest rates; see Section 5Â·2). By assuming that investors are risk-neutral, we have

assumed that the risk premium is zero. Investors are willing to take any investment that oï¬€ers

an expected rate of return of 10%, regardless of risk.

Although our example has been a little sterile, because we assumed away risk preferences, it is Unrealistic, maybe! But

ultimately, maybe not.

nevertheless very useful. Almost all projects in the real world are ï¬nanced with loans extended

by one party and levered ownership held by another party. Understanding debt and equity is

as important to corporations as it is to building owners. After all, stocks in corporations are

basically levered ownership claims that provide money only after the corporation has paid back

its loans. The building example has given you the skills to compute state-contingent, promised,

and expected payoï¬€s, and state-contingent, promised, and expected rates of returnsâ€”the nec-

essary tools to work with debt, equity, or any other state-contingent claim. And really, all that

will happen later when we introduce risk aversion is that we will add a couple of extra basis

ï¬le=uncertainty.tex: LP

100 Chapter 5. Uncertainty, Default, and Risk.

Figure 5.2. Probability Histograms of Project Returns

1.0

0.8

Histogram of Rate of Re-

0.6

Probability

turn for Project of Type

0.4

â€œFull Ownershipâ€

0.2

0.0

âˆ’100 âˆ’50 0 50 100

Rate of Return, in %

1.0

0.8

Histogram of Rate of Re-

0.6

Probability

turn for Project of Type

0.4

â€œLevered Ownershipâ€

0.2

(Most Risky)

0.0

âˆ’100 âˆ’50 0 50 100

Rate of Return, in %

1.0

0.8

Histogram of Rate of Re-

0.6

Probability

turn for Project of Type

0.4

â€œLoan Ownershipâ€

0.2

(Least Risky)

0.0

âˆ’100 âˆ’50 0 50 100

Rate of Return, in %

20% probability 80% probability Expected

Tornado Sunshine Average Variability

$20, 000 $29, 375 $27, 500

Loan (Ownership) âˆ’1= âˆ’1= âˆ’1=

$25, 000 $25, 000 $25, 000

âˆ’20.00% +17.50% â‰ˆ Â±20%

10.00%

$0 $70, 625 $56, 500

Levered Ownership âˆ’1= âˆ’1= âˆ’1=

$51, 364 $51, 364 $51, 364

âˆ’100.00% +37.50% â‰ˆ Â±60%

10.00%

$20, 000 $100, 000 $84, 000

Full Ownership âˆ’1= âˆ’1= âˆ’1=

$76, 364 $76, 364 $76, 364

âˆ’73.81% +30.95% â‰ˆ Â±46%

10.00%

Consider the variability number here to be just an intuitive measureâ€”it is the aforementioned â€œstandard deviationâ€

that will be explained in great detail in Chapter 13. Here, it is computed as

20% Â· (âˆ’20.00% âˆ’ 10%)2 + 80% Â· (17.50% âˆ’ 10%)2

Sdv = = 19.94%

Loan Ownership

20% Â· (âˆ’100.00% âˆ’ 10%)2 + 80% Â· (37.50% âˆ’ 10%)2 = 59.04%

Levered Ownership Sdv =

(5.37)

20% Â· (âˆ’73.81% âˆ’ 10%)2 + 80% Â· (30.95% âˆ’ 10%)2

Sdv = = 46.07%

Full Ownership

Prob T Â· [VT âˆ’ E (V )]2 + Prob S Â· [VS âˆ’ E (V )]2

Sdv = .

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101

Section 5Â·3. Uncertainty in Capital Budgeting, Debt, and Equity.

points of required compensationâ€”more to equity (the riskiest claim) than to the project (the

medium-risk claim) than to debt (the safest claim).

Solve Now!

Q 5.14 In the example, the building was worth $76,364, the mortgage was worth $25,000, and

the equity was worth $51,364. The mortgage thus ï¬nanced 32.7% of the cost of the building, and

the equity ï¬nanced $67.3%. Is the arrangement identical to one in which two partners purchase

the building togetherâ€”one puts in $25,000 and owns 32.7%, and the other puts in $51,364 and

owns 67.3%?

Q 5.15 Buildings are frequently ï¬nanced with a mortgage that pays 80% of the price, not just

32.7% ($25,000 of $76,364). Produce a table similar to Table 5.7 in this case.

Q 5.16 Repeat the example if the loan does not provide $25,000, but promises to pay oï¬€ $25,000.

How much money do you get for this promise? What is the promised rate of return. How does the

riskiness of the project â€œfull building ownershipâ€ compare to the riskiness of the project â€œlevered

building ownershipâ€?

Q 5.17 Repeat the example if the loan promises to pay oï¬€ $20,000. Such a loan is risk-free. How

does the riskiness of the project â€œfull building ownershipâ€ compare to the riskiness of the project

â€œlevered building ownershipâ€?

Q 5.18 Under risk-neutrality, a factory can be worth $500,000 or $1,000,000 in two years, de-

pending on product demand, each with equal probability. The appropriate cost of capital is 6%

per year. The factory can be ï¬nanced with proceeds of $500,000 from loans today. What are

the promised and expected cash ï¬‚ows and rates of return for the factory (without loan), for the

loan, and for a hypothetical factory owner who has to ï¬rst repay the loan?

Q 5.19 Advanced: For illustration, we assumed that the sample building was not lived in. It

consisted purely of capital amounts. But, in the real world, part of the return earned by a

building owner is rent. Now assume that rent of $11,000 is paid strictly at year-end, and that

both the state of nature (tornado or sun) and the mortgage loan payment happens only after the

rent has been safely collected. The new building has a resale value of $120,000 if the sun shines,

and a resale value of $20,000 if the tornado strikes.

(a) What is the value of the building today?

(b) What is the promised interest rate for a lender providing $25,000 in capital today?

(c) What is the value of residual ownership today?

(d) Conceptual Question: What is the value of the building if the owner chooses to live in the

building?

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102 Chapter 5. Uncertainty, Default, and Risk.

More Than Two Possible Outcomes

How does this example generalize to multiple possible outcomes? For example, assume that the

Multiple outcomes will

cause multiple building could be worth $20,000, $40,000, $60,000, $80,000, or $100,000 with equal probability,

breakpoints.

and the appropriate expected interest rate were 10%â€”so the building has an PV of $60, 000/(1+

10%) â‰ˆ $54, 545.45. If a loan promised $20,000, how much would you expect to receive? But,

of course, $20,000!

E Payoï¬€(Loan Promise =$20,000) = $20, 000 .

(5.38)

Payoï¬€ of Loan

E = .

Loan

if $0 â‰¤ Loan â‰¤ $20, 000

If a loan promised $20,001, how much would you expect to receive? $20,000 for sure, plus the

extra â€œmarginalâ€ $1 with 80% probability. In fact, you would expect only 80 cents for each dollar

promised between $20,000 and $40,000. So, if a loan promised $40,000, you would expect to

receive

E = $20, 000 + 80% Â· ($40, 000 âˆ’ $20, 000)

Payoï¬€( Loan Promise = $40,000 )

(5.39)

= $36, 000

Payoï¬€ of Loan

E = $20, 000 + 80% Â· (Loan âˆ’ $20, 000) .

if $20, 000 â‰¤ Loan â‰¤ $40, 000

If a loan promised you $40,001, how much would you expect to receive? You would get $20,000

for sure, plus another $20,000 with 80% probability (which is an expected $16,000), plus the

marginal $1 with only 60% probability. Thus,

E Payoï¬€(Loan Promise = $40,001) = + 80% Â· ($40, 000 âˆ’ $20, 000)

$20, 000

+ 60% Â· $1

(5.40)

= $36, 000.60

Payoï¬€ of Loan

E = + 80% Â· $20, 000

$20, 000

if $40, 000 â‰¤ Loan â‰¤ $60, 000

+ 60% Â· (Loan âˆ’ $40, 000) .

And so on. Figure 5.3 plots these expected payoï¬€s as a function of the promised payoï¬€s. With

this ï¬gure, mortgage valuation becomes easy. For example, how much would the loan have to

promise to provide $35,000 today? The expected payoï¬€ would have to be (1+10%)Â·$35, 000 =

$38, 500. Figure 5.3 shows that an expected payoï¬€ of $38,500 corresponds to around $44,000

in promise. (The exact number can be worked out to be $44,167.) Of course, we cannot borrow

more than $54,545.45, the projectâ€™s PV. So, we can forget about the idea of obtaining a $55,000

mortgage.

Solve Now!

Q 5.20 What is the expected payoï¬€ if the promised payoï¬€ is $45,000?

Q 5.21 What is the promised payoï¬€ if the expected payoï¬€ is $45,000?

Q 5.22 Assume that the probabilities are not equal: $20,000 with probability 12.5%, $40,000

with probability 37.5%, $60,000 with probability 37.5%, and $80,000 with probability 12.5%.

(a) Draw a graph equivalent to Figure 5.3.

(b) If the promised payoï¬€ of a loan is $50,000, what is the expected payoï¬€?

(c) If the prevailing interest rate is 5% before loan payoï¬€, then how much repayment does a

loan providing $25,000 today have to promise? What is the interest rate?

You do not need to calculate these values, if you can read them oï¬€ your graph.

ï¬le=uncertainty.tex: RP

103

Section 5Â·3. Uncertainty in Capital Budgeting, Debt, and Equity.

Figure 5.3. Promised vs. Expected Payoï¬€s

60

50

40

Expect

30

20

10

0

0 20 40 60 80 100

Promise

Q 5.23 A new product may be a dud (20% probability), an average seller (70% probability) or

dynamite (10% probability). If it is a dud, the payoï¬€ will be $20,000; if it is an average seller, the

payoï¬€ will be $40,000; if it is dynamite, the payoï¬€ will be $80,000. The appropriate expected

rate of return is 6% per year. If a loan promises to pay oï¬€ $40,000, what are the promised and

expected rates of return?

Q 5.24 Advanced: What is the formula equivalent to (5.40) for promised payoï¬€s between $60,000

and $80,000?

Q 5.25 Advanced: Can you work out the exact $44,167 promise for the $35,000 (today!) loan?

ï¬le=uncertainty.tex: LP

104 Chapter 5. Uncertainty, Default, and Risk.

5Â·4. Robustness: How Bad are Your Mistakes?

Although it would be better to get everything perfect, it is often impossible to come up with

How bad are mistakes?

perfect cash ï¬‚ow forecasts and appropriate interest rate estimates. Everyone makes errors.

So, how bad are mistakes? How robust is the NPV formula? Is it worse to commit an error in

estimating cash ï¬‚ows or in estimating the cost of capital? To answer these questions, we will

do a simple form of scenario analysisâ€”we will consider just a very simple project, and see

how changes in estimates matter to the ultimate value. Doing good scenario analysis is also

good practice for any managersâ€”so that they can see how sensitive their estimated value is to

reasonable alternative possible outcomes. Therefore this method is also called a sensitivity

analysis. Doing such analysis becomes even more important when we consider â€œreal optionsâ€

in our next chapter.

5Â·4.A. Short-Term Projects

Assume that your project will pay oï¬€ $200 next year, and the proper interest rate for such

The benchmark case: A

short-term project, projects is 8%. Thus, the correct project present value is

correctly valued.

$200

(5.44)

PVcorrect = â‰ˆ $185.19 .

1 + 8%

If you make a 10% error in your cash ï¬‚ow, e.g., mistakenly believing it to return $220, you will

Committing an error in

cash ï¬‚ow estimation. compute the present value to be

$220

(5.45)

PVCF error = â‰ˆ $203.70 .

1 + 8%

The diï¬€erence between $203.70 and $185.19 is a 10% error in your present value.

In contrast, if you make a 10% error in your cost of capital (interest rate), mistakenly believing

Committing an error in

interest rate estimation. it to require a cost of capital (expected interest rate) of 8.8% rather than 8%, you will compute

the present value to be

$200

(5.46)

= â‰ˆ $183.82 .

PVr error

1 + 8.8%

The diï¬€erence between $183.82 and $185.19 is less than a $2 or 1% error.

Important: For short-term projects, errors in estimating correct interest rates

are less problematic in computing NPV than are errors in estimating future cash

ï¬‚ows.

5Â·4.B. Long-Term Projects

Now take the same example, but assume the cash ï¬‚ow will occur in 30 years. The correct

A long-term project,

correctly valued and present value is now

incorrectly valued.

$200

PVcorrect = â‰ˆ $19.88 (5.47)

.

(1 + 8%)30

The 10% â€œcash ï¬‚ow errorâ€ present value is

$220

PVCF error = â‰ˆ $21.86 (5.48)

,

(1 + 8%)30

and the 10% â€œinterest rate errorâ€ present value is

$200

= â‰ˆ $15.93 (5.49)

.

PVr error

(1 + 8.8%)30

ï¬le=uncertainty.tex: RP

105

Section 5Â·4. Robustness: How Bad are Your Mistakes?.

This calculation shows that cash ï¬‚ow estimation errors and interest rate estimation errors are Both cash ï¬‚ow and cost

of capital errors are now

now both important. So, for longer-term projects, estimating the correct interest rate becomes

important.

relatively more important. However, in fairness, estimating cash ï¬‚ows thirty years into the

future is just about as diï¬ƒcult as reading a crystal ball. (In contrast, the uncertainty about the

long-term cost of capital tends not explode as quickly as your uncertainty about cash ï¬‚ows.)

Of course, as diï¬ƒcult as it may be, we have no alternative. We must simply try to do our best

at forecasting.

Important: For long-term projects, errors in estimating correct interest rates

and errors in estimating future cash ï¬‚ows are both problematic in computing NPV.

5Â·4.C. Two Wrongs Do Not Make One Right

Please do not think that you can arbitrarily adjust an expected cash ï¬‚ow to paint over an issue Do not think two errors

cancel.

with your discount rate estimation, or vice-versa. For example, let us presume that you consider

an investment project that is a bond issued by someone else. You know the bondâ€™s promised

payoï¬€s. You know these are higher than the expected cash ï¬‚ows. Maybe you can simply use

the average promised discount rate on other risky bonds to discount the bondâ€™s promised cash

ï¬‚ows? After all, the latter also reï¬‚ects default risk. The two default issues might cancel one

another, and you might end up with the correct number. Or they might not cancel and you end

up with a non-sense number!

Letâ€™s say the appropriate expected rate of return is 10%. A suggested bond investment may An example: do not

think you can just work

promise $16,000 for a $100,000 investment, but have a default risk on the interest of 50% (the

with promised rates.

principal is insured). Your benchmark promised opportunity cost of capital may rely on risky

bonds that have default premia of 2%. Your project NPV is neither âˆ’$100, 000 + $116, 000/(1 +

12%) â‰ˆ $3, 571 nor âˆ’$100, 000 + $100, 000/(1 + 10%) + $16, 000/(1 + 12%) â‰ˆ $5, 195. Instead,

you must work with expected values

$100, 000 $8, 000

(5.50)

PV = âˆ’$100, 000 + + â‰ˆ âˆ’$1, 828 .

1 + 10% 1 + 10%

This bond would be a bad investment.

Solve Now!

Q 5.26 What is the relative importance of cash ï¬‚ow and interest rate errors for a 10-year project?

Q 5.27 What is the relative importance of cash ï¬‚ow and interest rate errors for a 100-year

project?

ï¬le=uncertainty.tex: LP

106 Chapter 5. Uncertainty, Default, and Risk.

5Â·5. Summary

The chapter covered the following major points:

â€¢ The possibility of future default causes promised interest rates to be higher than expected

interest rates. Default risk is also often called credit risk.

â€¢ Quoted interest rates are almost always promised interest rates, and are higher than

expected interest rates.

â€¢ Most of the diï¬€erence between promised and expected interest rates is due to default.

Extra compensation for bearing more riskâ€”the risk premiumâ€”is typically much smaller

than the default premium.

â€¢ The key tool for thinking about uncertainty is the payoï¬€ table. Each row represents one

possible state outcome, which contains the probability that the state will come about, the

total project value that can be distributed, and the allocation of this total project value

to diï¬€erent state-contingent claims. The state-contingent claims â€œcarve upâ€ the possible

project payoï¬€s.

â€¢ Most real-world projects are ï¬nanced with the two most common state-contingent claimsâ€”

debt and equity. The conceptual basis of debt and equity is ï¬rmly grounded in payoï¬€

tables. Debt ï¬nancing is the safer investment. Equity ï¬nancing is the riskier investment.

â€¢ If debt promises to pay more than the project can deliver in the worst state of nature, then

the debt is risky and requires a promised interest rate in excess of its expected interest

rate.

â€¢ NPV is robust to uncertainty about the expected interest rate (the discount rate) for short-

term projects. However, NPV is not robust with respect to either expected cash ï¬‚ows or

discount rates for long-run projects.

ï¬le=uncertainty.tex: RP

107

Section 5Â·5. Summary.

Solutions and Exercises

1. No! It is presumed to be knownâ€”at least for a die throw. The following is almost philosophy and beyond

what you are supposed to know or answer here: It might, however, be that the expected value of an investment

is not really known. In this case, it, too, could be a random variable in one senseâ€”although you are presumed

to be able to form an expectation (opinion) over anything, so in this sense, it would not be a random variable,

either.

2. Yes and no. If you do not know the exact bet, you may not know the expected value.

3. If the random variable is the number of dots on the die, then the expected outcome is 3.5. The realization

was 6.

4. The expected value of the stock is $52. Therefore, purchasing the stock at $50 is not a fair bet, but a good

bet.

5. Only for government bonds. Most other bonds have some kind of default risk.

6.

(a) The expected payoï¬€ is now 95%Â·$210 + 1%Â·$100 + 4%Â·$0 = $200.50. Therefore, the expected rate of

return is $200.50/$200 = 0.25%.

(b) You require an expected payoï¬€ of $210. Therefore, you must solve for a promised payment 95%Â·P +

1%Â·$100 + 4%Â·$0 = $210 â†’ P = $209/0.95 = $220. On a loan of $200, this is a 10% promised interest

rate.

7. The expected payoï¬€ is $203.50, the promised payoï¬€ is $210, and the stated price is $210/(1+12%)=$187.50.

The expected rate of return is $203.50/$187.50 = 8.5%. Given that the time premium, the Treasury rate is 5%,

the risk premium is 3.5%. The remaining 12%-8.5%=3.5% is the default premium.

8. The actual cash ï¬‚ow is replaced by the expected cash ï¬‚ow, and the actual rate of return is replaced by the

expected rate of return.

9. $750, 000/(1 + 6%)2 â‰ˆ $667, 497.33.

10. E (P ) = 20% Â· $20, 000 + 70% Â· $40, 000 + 10% Â· $80, 000 = $40, 000.

11. $37, 037.

12. $20, 000/$37, 037 âˆ’ 1 = âˆ’46%, $40, 000/$37, 037 âˆ’ 1 = +8%, $80, 0000/$37037 âˆ’ 1 = +116%.

13. 20%Â·(âˆ’46%) + 70%Â·(+8%) + 10%Â·(+116%) = 8%.

14. No! Partners would share payoï¬€s proportionally, not according to â€œdebt comes ï¬rst.â€ For example, in the

tornado state, the 32.7% partner would receive only $6,547.50, not the entire $20,000 that the debt owner

receives.

15. The mortgage would ï¬nance $61,090.91 today.

Prob

Event Building Value Mortgage Value Levered Ownership

Tornado 20% $20,000 $20,000 $0

Sunshine 80% $100,000 $79,000 $21,000

Expected Valuet=1 $84,000 $67,200 $16,800

PVt=0 $76,364 $61,091 $15,273

E (rt=0,1 ) 10% 10% 10%

16. In the tornado state, the creditor gets all ($20,000). In the sunshine state, the creditor receives the promise

of $25, 000. Therefore, the creditorâ€™s expected payoï¬€ is 20% Â· $20, 000 + 80% Â· $25, 000 = $24, 000. To oï¬€er

an expected rate of return of 10%, you can get $24, 000/1.1 = $21, 818 from the creditor today. The promised

rate of return is therefore $25, 000/$21, 818 âˆ’ 1 = 14.58%.

17. The loan pays oï¬€ $20,000 for certain. The levered ownership pays either $0 or $80,000, and costs $64, 000/(1+

10%) = $58, 182. Therefore, the rate of return is either âˆ’100% or +37.5%. We have already worked out full

ownership. It pays either $20,000 or $100,000, costs $76,364, and oï¬€ers either âˆ’73.81% or +30.95%. By

inspection, the levered equity project is riskier. In eï¬€ect, building ownership has become riskier, because the

owner has chosen to sell oï¬€ the risk-free component, and retain only the risky component.

18. Factory: The expected factory value is $750,000. Its price would be $750, 000/1.062 = $667, 497. The

promised rate of return is therefore $1, 000, 000/$667, 497 âˆ’ 1 â‰ˆ 49.8%. Loan: The discounted (todayâ€™s) loan

price is $750, 000/1.062 = $667, 497.33. The promised value is $1,000,000. The loan must have an expected

payoï¬€ of 1.062 Â· $500, 000 = $561, 800 (6% expected rate of return, two years). Because the loan can pay

$500,000 with probability 1/2, it must pay $623,600 with probability 1/2 to reach $561,800 as an average.

Therefore, the promised loan rate of return is $623, 600/$500, 000 âˆ’ 1 = 24.72% over two years (11.68%

per annum). Equity: The levered equity must therefore pay for/be worth $667, 497.33 âˆ’ $500, 000.00 =

$167, 497.33 (alternatively, levered equity will receive $1, 000, 000 âˆ’ $623, 600 = $376, 400 with probability

1/2 and $0 with probability 1/2), for an expected payoï¬€ of $188,200. (The expected two-year holding rate

of return is $188, 200/$167, 497 âˆ’ 1 = 12.36% [6% per annum, expected].) The promised rate of return is

($1, 000, 000 âˆ’ $623, 600)/$167, 497.33 âˆ’ 1 = 124.72% (50% promised per annum).

ï¬le=uncertainty.tex: LP

108 Chapter 5. Uncertainty, Default, and Risk.

19.

(a) In the sun state, the value is $120,000+$11,000= $131,000. In the tornado state, the value is $11,000+$20,000=

$31,000. Therefore, the expected building value is $111,000. The discounted building value today is

$100,909.09.

(b) Still the same as in the text: the lenderâ€™s $25,000 loan can still only get $20,000, so it is a promise for

$29,375. So the quoted interest rate is still 17.50%.

(c) $100, 909.09 âˆ’ $25, 000 = $75, 909.09.

(d) Still $100,909.09, assuming that the owner values living in the building as much as a tenant would.

Owner-consumed rent is the equivalent of corporate dividends paid out to levered equity. Note: you can repeat

this example assuming that the rent is an annuity of $1,000 each month, and tornadoes strike mid-year.

20. From the graph, it is around $40,000. The correct value can be obtained by plugging into Formula (5.40):

$39,000.

21. From the graph, it is around $55,000. The correct value can be obtained by setting Formula (5.40) equal to

$55,000 and solving for â€œLoan.â€ The answer is indeed $55,000.

22.

60

50

40

Expect

30

20

10

0

0 20 40 60 80 100

Promise

(a)

(b) The exact expected payoï¬€ is 1/8 Â· $20, 000 + 3/8 Â· $40, 000 + 1/2 Â· $50, 000 = $42, 500. The 1/2 is the

probability that you will receive the $50,000 that you have been promised, which occurs if the project

ends up worth at least as much as your promised $50,000. This means that it is the total probability

that it will be worth $60,000 or $80,000.

(c) The loan must expect to pay oï¬€ (1 + 5%) Â· $25, 000 = $26, 250. Therefore, solve 1/8 Â· $20, 000 + 7/8 Â· x =

$26, 250, so the exact promised payoï¬€ must be x = $27, 142.90.

23. With 20% probability, the loan will pay oï¬€ $20,000; with 80% probability, the loan will pay oï¬€ the full promised

$40,000. Therefore, the loanâ€™s expected payoï¬€ is 20%Â·$20, 000 + 80%Â·$40, 000 = $36, 000. The loanâ€™s price

is $36, 000/(1 + 6%) = $33, 962. Therefore, the promised rate of return is $40, 000/$33, 962 âˆ’ 1 â‰ˆ 17.8%. The

expected rate of return was given: 6%.

24.

Payoï¬€ of Loan

E = $20, 000

if $60, 000 â‰¤ Loan â‰¤ $80, 000

+ 80% Â· $20, 000

(5.51)

+ 60% Â· $20, 000

+ 40% Â· (Loan âˆ’ $60, 000) .

25. The loan must yield an expected value of $38,500. Set formula (5.40) equal to $38,500 and solve for â€œLoan.â€

The answer is indeed $44,166.67.

ï¬le=uncertainty.tex: RP

109

Section 5Â·5. Summary.

26. Consider a project that earns $100 in 10 years, and where the correct interest rate is 10%.

â€¢ The correct PV is $100/(1 + 10%)10 = $38.55.

â€¢ If the cash ï¬‚ow is incorrectly estimated to be 10% higher, the incorrect PV is $110/(1 + 10%)10 = $42.41.

â€¢ If the interest rate is incorrectly estimate to be 10% lower, the incorrect PV is $100/(1 + 9%)10 = $42.24.

So, the misvaluation eï¬€ects are reasonably similar at 10% interest rates. Naturally, percent valuation mistakes

in interest rates are higher if the interest rate is higher; and lower if the interest rate is lower.

27. Although this, too, depends on the interest rate, interest rate errors almost surely matter for any reasonable

interest rates now.

(All answers should be treated as suspect. They have only been sketched, and not been checked.)

ï¬le=uncertainty.tex: LP

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